A modular functor which is universal for quantum computation

TL;DR

This paper demonstrates that a topological modular functor derived from Witten-Chern-Simons theory can perform universal quantum computation, establishing a connection between topological quantum field theory and quantum computing models.

Contribution

It introduces a computational model based on Chern-Simons theory at a fifth root of unity and proves its polynomial equivalence to the standard quantum circuit model.

Findings

The topological modular functor is universal for quantum computation.

A new quantum computational model based on Chern-Simons theory is defined.

The model is polynomially equivalent to the quantum circuit model.

Abstract

We show that the topological modular functor from Witten-Chern-Simons theory is universal for quantum computation in the sense a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor's state space. A computational model based on Chern-Simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model. The chief technical advance: the density of the irreducible sectors of the Jones representation, have topological implications which will be considered elsewhere.